### Abstract

Subject of investigation in this paper is a one-dimensional Schrödinger equation, where the potential is a sum of a periodic function and a perturbation decaying at ±∞. It is well known that the essential spectrum consists of spectral bands, and that there may or may not be additional eigenvalues below the lowest band or in the gaps between the bands. While enclosures for gap eigenvalues can comparatively easily be obtained from numerical approximations, e.g. by D. Weinstein's bounds, there seems to be no method available so far which is able to exclude eigenvalues in spectral gaps, i.e. which identifies subregions (of a gap) which contain no eigenvalues. Here, we propose such a method. It makes heavy use of computer assistance; nevertheless, the results are completely rigorous in the strict mathematical sense, because all computational errors are taken into account.

Original language | English |
---|---|

Pages (from-to) | 545-562 |

Number of pages | 18 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 468 |

Issue number | 2138 |

DOIs | |

Publication status | Published - 2012 Feb 8 |

Externally published | Yes |

### Fingerprint

### Keywords

- Eigenvalue excluding
- Error estimates
- Schrödinger operator

### ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)
- Physics and Astronomy(all)

### Cite this

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*,

*468*(2138), 545-562. https://doi.org/10.1098/rspa.2011.0159

**Eigenvalue excluding for perturbed-periodic one-dimensional Schrödinger operators.** / Nagatou, Kaori; Plum, Michael; Nakao, Mitsuhiro T.

Research output: Contribution to journal › Article

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, vol. 468, no. 2138, pp. 545-562. https://doi.org/10.1098/rspa.2011.0159

}

TY - JOUR

T1 - Eigenvalue excluding for perturbed-periodic one-dimensional Schrödinger operators

AU - Nagatou, Kaori

AU - Plum, Michael

AU - Nakao, Mitsuhiro T.

PY - 2012/2/8

Y1 - 2012/2/8

N2 - Subject of investigation in this paper is a one-dimensional Schrödinger equation, where the potential is a sum of a periodic function and a perturbation decaying at ±∞. It is well known that the essential spectrum consists of spectral bands, and that there may or may not be additional eigenvalues below the lowest band or in the gaps between the bands. While enclosures for gap eigenvalues can comparatively easily be obtained from numerical approximations, e.g. by D. Weinstein's bounds, there seems to be no method available so far which is able to exclude eigenvalues in spectral gaps, i.e. which identifies subregions (of a gap) which contain no eigenvalues. Here, we propose such a method. It makes heavy use of computer assistance; nevertheless, the results are completely rigorous in the strict mathematical sense, because all computational errors are taken into account.

AB - Subject of investigation in this paper is a one-dimensional Schrödinger equation, where the potential is a sum of a periodic function and a perturbation decaying at ±∞. It is well known that the essential spectrum consists of spectral bands, and that there may or may not be additional eigenvalues below the lowest band or in the gaps between the bands. While enclosures for gap eigenvalues can comparatively easily be obtained from numerical approximations, e.g. by D. Weinstein's bounds, there seems to be no method available so far which is able to exclude eigenvalues in spectral gaps, i.e. which identifies subregions (of a gap) which contain no eigenvalues. Here, we propose such a method. It makes heavy use of computer assistance; nevertheless, the results are completely rigorous in the strict mathematical sense, because all computational errors are taken into account.

KW - Eigenvalue excluding

KW - Error estimates

KW - Schrödinger operator

UR - http://www.scopus.com/inward/record.url?scp=84857208536&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84857208536&partnerID=8YFLogxK

U2 - 10.1098/rspa.2011.0159

DO - 10.1098/rspa.2011.0159

M3 - Article

VL - 468

SP - 545

EP - 562

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 0080-4630

IS - 2138

ER -